Mathematical modeling

In a given period there are 600 hours of machine A time available, 850 hours ofmachine B available, and 800 hours of machine C. The expected profit for theproduction of each unit package is LE 2000 for type 1 and LE 1900 for type 2.Formulate a linear program model that maximizes the profit and determine thenumber of house type produces.9. A Corporation produces three types of sprinkler heads for irrigation applications.The company currently has orders for 600 two-inch heads, 500 three-inch heads,and 800 four-inch heads. Sprinkler heads are made from a copper alloy using amold injection process. The company has four different models of molds availablefor use. The first mold, x 1 , produces 15 two-inch heads per pound of alloy; thesecond, x 2 , produces 20 three-inch heads per pound; the third, x 3 , produces 14three-inch heads and 40 four-inch heads per pound; and the fourth, x 4 , produces10 two-inch heads and 20 four-inch heads per pound. The production managerwants to determine the optimal production method to minimize the amount ofcopper used to meet demand. Formulate the problem as a linear programmingmodel.10. For the following constraint set find:x 1 + x 2 ≤ 3 (1)1 ≤ x 2 ≤ 2 (2)x 1 ≥ 0 (3)a. The feasible region on a graph for the constraint set.b. Label the extreme points.c. If the less than or equal to constraint of Eq. (1) is changed to a strict equality,show the feasible region and label the extreme points.11. Pharma produces two chemicals: A and B. These chemicals are produced via twomanufacturing processes. Process 1 requires 2 hours of labor and 1 lb. of rawmaterial to produce 2 oz. of A and 1 oz. of B. Process 2 requires 3 hours of laborand 2 lb. of raw material to produce 3 oz. of A and 2 oz. of B. Sixty hours of laborand 40 lb. of raw material are available. Chemical A sells for LE 16 per oz. and BSystems Analysis 47 Dr. Emad Elbeltagi